We consider in this paper distributed hypothesis testing (DHT) with a single sensor Alice and a single decision center Bob, each observing an independently and identically distributed (i.i.d.) source sequence, where the two sequences are jointly drawn according to the known probability mass function \(P_{XY}\)  under hypothesis \(\mathcal {H}_0\)  and according to the product of the marginals \(P_X P_Y\)  under \(\mathcal {H}_1\) . Information-theorists refer to this setup as testing against independence. Alice can send a rate-\(R\)  message to Bob describing her observations and aiming to help Bob in deciding on the true hypothesis. The focus here is on the Stein exponent, i.e.,
on the largest possible exponential decay for Bob's type-II error probability under the requirement that his type-I error probability stays below a given threshold \(\epsilon \in (0,1)\) . This largest possible type-II error exponent in this setup was determined by Ahlswede and Csiszár [1]} and does not depend on the value of \(\epsilon \) . In this paper, we consider an extension of the Ahlswede-Csiszár result to a setup including an additional eavesdropper Eve that observes a local i.i.d. source sequence, intercepts Alice's message to Bob \(M\) , and wishes to learn about Alice's source sequence \(X^n\) . In this extended setup, Alice is required to choose her message in a way that Eve's equivocation about the source \(X^n\)  stays above pre-determined thresholds given the two hypothesis.
